Chapter 3: Continuum Mechanics
Vascular Biomechanics by T. Christian Gasser (2022)
Reading Notes
3.1 Introduction
Continuum mechanics provides a localized approach to analyze vascular problems, complementing the lumped parameter models from previous chapters. The method introduces the Representative Volume Element (RVE) — a volume large enough to homogenize material properties but small relative to the characteristic dimension of the problem.
Key concepts: - RVE (Representative Volume Element): Bridges microstructural heterogeneity and macrostructural analysis - Two-scale separation: global scale (FE discretization) and local scale (homogenized via RVE) - Vascular tissue exhibits hierarchical structure that may challenge this scale separation
3.2 Kinematics
Kinematics describes the deformation of a body from reference configuration Ω₀ to spatial configuration Ω.
3.2.1 Deformation Gradient
F(X, t) = Gradχ(X, t) = ∂χ(X, t)/∂X
- Two-point tensor connecting reference and spatial configurations
- detF > 0 prevents self-penetration
- Eigenvalue representation: F = Σ λᵢ nᵢ ⊗ Nᵢ (principal stretches λᵢ)
3.2.2 Multiplicative Decomposition
F(X) = FₙFₙ₋₁⋯F₁ — total deformation gradient as product of incremental gradients
3.2.3 Polar Decomposition
- Right polar: F = RU (U = right stretch tensor)
- Left polar: F = vQ (v = left stretch tensor)
- R, Q = rotation tensors (orthogonal)
3.2.4 Line Element Deformation
λₐ = dl/dL = √(a₀ · C · a₀) = |Fa₀|
Where C = FᵀF is the right Cauchy-Green strain, a₀ is the fiber direction vector.
3.2.5 Volume Element Deformation
dv = J dV (J = detF = volume ratio)
3.2.6 Area Element Deformation (Nanson's Formula)
ds = J F⁻ᵀ dS or dsi = J F⁻¹ᵃᵢ dSₐ
3.2.7 Concept of Strain
Engineering (Linear) Strain (small deformations, Ω₀ ≈ Ω): - ε = ½(grad u + (grad u)ᵀ) - Component form: εᵢⱼ = (∂uᵢ/∂xⱼ + ∂uⱼ/∂xᵢ)/2
Non-linear Strain Measures:
| Strain Measure | Definition | Configuration |
|---|---|---|
| Right Cauchy-Green | C = FᵀF | Reference Ω₀ |
| Left Cauchy-Green | b = FFᵀ | Spatial Ω |
| Green-Lagrange | E = ½(C - I) | Reference Ω₀ |
| Euler-Almansi | e = ½(I - b⁻¹) | Spatial Ω |
Properties: - C and b are relative strain measures (C = b = I when strain-free) - E and e are absolute strain measures (E = e = 0 when strain-free) - At small strains: E → ε, e → ε when Ω₀ ≈ Ω
3.3 Concept of Stress
Stress represents force per unit area acting on a material particle.
3.3.1 Cauchy Stress Theorem
t(x) = σ(x)n(x) or tᵢ = σᵢₐ nₐ
Traction vector t acts on area element ds with unit normal n.
3.3.2 Principal Stresses
Eigenvalue problem: (σ - λI)n = 0
Stress Invariants: - I₁ = trσ = σ₁₁ + σ₂₂ + σ₃₃ - I₂ = ½[(trσ)² - trσ²] - I₃ = detσ
3.3.3 Mohr's Circle
Coordinate rotation in physical space (angle α) corresponds to counter-clockwise rotation (2α) in stress space.
3.3.4 Isochoric and Volumetric Stress
σ = σ̄ + σᵛᵒˡ = σ̄ - pI
Where: - σ̄ = devσ (deviatoric/isochoric stress) - σᵛᵒˡ = -pI (hydrostatic stress) - p = -trσ/ndim
3.3.5 Octahedral and von Mises Stress
Octahedral normal stress: σₒcₜ = I₁/3 = -p Octahedral shear stress: τₒcₜ = √[½((σ₁-σ₂)² + (σ₂-σ₃)² + (σ₁-σ₃)²)] = √(-⅔J₂)
von Mises Stress: σₘ = √3τₒcₜ = √(-3J₂)
Or in component form: σₘ = ½√[(σ₁₁-σ₂₂)² + (σ₂₂-σ₃₃)² + (σ₁₁-σ₃₃)² + 6(σ₁₂² + σ₂₃² + σ₁₃²)]
3.3.7 First Piola-Kirchhoff Stress (P)
P = JσF⁻ᵀ or Pᵢᵢ = JσᵢₐFᵢₐ⁻¹
- Two-point tensor
- Non-symmetric when represented as matrix
- Transforms as: P̃ = RP
3.3.8 Second Piola-Kirchhoff Stress (S)
S = JF⁻¹σF⁻ᵀ or Sᵢⱼ = JFᵢₐ⁻¹σₐᵦFJᵦ⁻¹
- Referential one-point tensor (defined in Ω₀)
- Symmetric: S = Sᵀ
- Not affected by rigid body rotation
- Relation: S = F⁻¹P
3.3.9 Incompressibility Constraint
For incompressible material: S = S - pC⁻¹
3.4 Material Time Derivatives
3.4.1 Kinematic Variables
Velocity Gradient: l = grad v = ḞF⁻¹
Rate of Deformation Tensor: d = ½(F⁻ᵀḞ + ḞF⁻¹) = ½(l + lᵀ)
Spin Tensor: w = ½(l - lᵀ) = ½(ḞF⁻¹ - F⁻ᵀḞᵀ)
3.4.2 Stress Rates
- Time derivative of Cauchy stress σ̇ is NOT objective
- Second Piola-Kirchhoff stress rate Ṡ is objective (a priori)
- Truesdell rate: σₜ = J⁻¹FṠFᵀ
- Jaumann rate: σⱼ = σ̇ + σw - wσ
3.4.3 Power-Conjugate Stress/Strain Rates
ṁ = S : Ċ/2 = S : Ė = P : Ḟ
- Second Piola-Kirchhoff stress S is power-conjugate to Ċ/2 and Ė
- First Piola-Kirchhoff stress P is power-conjugate to Ḟ
- Cauchy stress σ is power-conjugate to rate of deformation d
3.5 Constitutive Modeling
3.5.1 Material Properties
- Incompressibility: Volume preserved; volumetric stress independent of deformation
- Isotropy/Anisotropy: Property dependence on spatial orientation
- Strain Energy: Work stored during deformation
- Dissipation: Energy converted to heat (viscoelastic materials)
- Stiffness: Fourth-order tensor C relating stress and strain increments
- Strength: Stress level at mechanical failure
3.5.2 Linear-Elastic (Hookean) Material
σ = Eε (1D)
Relation between moduli: E = 2(1+ν)G
Where E = Young's modulus, ν = Poisson's ratio, G = shear modulus
Hooke's Law (Voigt notation) — 6×6 compliance matrix relating ε and σ
Tensor form: cᵢⱼₖₗ = [Eν/((1+ν)(1-2ν))]δᵢⱼδₖₗ + E/(2(1+ν))
3.5.3 Hyperelasticity
Assumes deformation energy is fully recovered upon unloading (no dissipation).
Helmholtz Free Energy: ψ = U - θS [J/m³]
Coupled Formulation: ψ = ψ(C) where C = right Cauchy-Green strain S = 2∂ψ/∂C
Volumetric-Isochoric Decoupled: ψ(J, C) = ψᵢₛₒ(C) + ψᵥₒₗ(J)
For incompressible: J = 1, ψᵥₒₗ "degenerates" to Lagrange multiplier
Second Piola-Kirchhoff stress (incompressible): S = 2Dev[∂ψᵢₛₒ/∂C] - κC⁻¹
3.5.4 Viscoelasticity
Work not entirely recovered upon unloading.
Newtonian Fluid: σ = 2ηd - pI (η = dynamic viscosity)
Linear Viscoelastic (Boltzmann superposition): - Creep: ε(t) = ∫ J(t-ξ)σ̇(ξ)dξ - Relaxation: σ(t) = ∫ G(t-ξ)ε̇(ξ)dξ
Maxwell Element (spring + dashpot in series): ε̇ = σ̇/E + σ/η Relaxation: σ(t) = σ₀exp(-t/τ) where τ = η/E
Kelvin-Voigt Element (spring + dashpot in parallel): σ = Eε + ηε̇
Standard Solid Element (Maxwell + spring in parallel): σ = σₘ + σₑ
3.5.5 Multiphasic Continuum Theories
Mixture Theory: n interpenetrable continua coexisting at material point ψ(C) = Σ ξᵢψᵢ(C) with Σξᵢ = 1
Poroelasticity: Solid skeleton + fluid in separate RVE domains Biot decomposition: σ = σᵉ - αpI
3.6 Governing Laws
3.6.1 Mass Balance
Dm/Dt = 0 (mass of material particle constant)
Lagrangian form: ∂ρ/∂t + ρdivv = 0 Incompressible: divv = 0
3.6.2 Balance of Linear Momentum
Cauchy's equation: ρDv/Dt = divσ + bƒ
Lagrangian: Dv/Dt = ∂v/∂t Eulerian: Dv/Dt = ∂v/∂t + gradv·v (includes advective acceleration)
Navier-Stokes Equation (Newtonian fluid): ρ(∂v/∂t + v·gradv) = ηdiv(gradv) - gradp + bƒ
3.6.3 Thermodynamic Laws
First Law: ė = hᵢₙₚᵤₜ + pᵢₙₚᵤₜ (energy conserved) Second Law: γ = ṡ - rh/θ - div(qh/θ) ≥ 0 (entropy produced)
Clausius-Duhem Inequality (isothermal): -ψ̇ + σ : d ≥ 0
3.6.4 Stress-Helmholtz Free Energy Relation
Coleman-Noll Procedure:
For compressible: P = ∂ψ/∂F or S = 2∂ψ/∂C
For incompressible (Lagrange multiplier κ): S = 2∂ψᵢₛₒ/∂C - κC⁻¹
Note: κ ≠ hydrostatic pressure p
3.7 General Principles
3.7.1 Maxwell Transport and Localization
Two-step derivation of balance equations: 1. Pull-back to reference configuration 2. Push-forward to spatial configuration, then localize
3.7.2 Free Body Diagram (FBD)
Abstraction of forces for equilibrium analysis: - Internal forces from hypothetical sectioning - Relate to stress via Euler's first and second principles - Example: Thin-walled sphere inflation σ = rpi/(2h)
3.7.3 Boundary Value Problem (BVP)
Mathematical description of continuum mechanics problem:
Strong form (solid mechanics): - Momentum: divσ + bƒ = ρD²u/Dt² in Ω - Dirichlet BC: u = ū on ∂Ωᵤ - Neumann BC: t = t̄ on ∂Ωₜ
Strong form (fluid mechanics): - Momentum: divσ + bƒ = ρDv/Dt in Ω - Velocity BC on ∂Ωᵥ - Traction BC on ∂Ωₜ
3.7.4 Principle of Virtual Work (PVW)
δWₑₓₜ = δWᵢₙₜ
Small deformation (static): ∫σ : δε dV = ∫bƒ · δu dV + ∫t · δu ds
Finite deformation (material form): ∫S : δE dV = ∫Bƒ · δu dV + ∫T · δu dS
3.8 Damage and Failure
3.8.1 Damage Mechanics
Isotropic damage: σ = (1-D)σᵉᶠᶠ (D = scalar damage parameter) Anisotropic damage: Higher-order damage tensor
Consequence: strain softening (stress decreases with increasing strain) - Stiffness tensor no longer positive definite - Problem changes from elliptic to hyperbolic
3.8.2 Strain Localization
- Deformation localizes in narrow domain
- Non-polar continuum requires regularization
- Gradient-enhanced damage models introduce internal length scale
3.8.3 Linear Fracture Mechanics (LFM)
Stress intensity factors: Kᵢ = lim√(2πx₁)σ (i = I, II, III) Energy release rate: D = f(Kᵢ, E, ν)
Limitations: Small strains required; sharp crack tip assumption
3.8.4 Non-linear Fracture Mechanics
J-Integral: Path-independent energy release rate calculation Cohesive Zone Modeling: Traction-Separation Law (TSL) as failure surrogate
3.9 Summary
Continuum mechanics enables exploration of both solid and fluid mechanical aspects of vascular circulation. The RVE represents material microstructure, while stress emerges as the most fundamental quantity. Constitutive models express stress-strain relationships, with complexity ranging from linear elasticity to viscohyperelasticity with multiple internal variables. The initial Boundary Value Problem (iBVP) mathematically frames biomechanical questions, typically requiring numerical approximation (e.g., FEM) for solution.
Formula Table (公式汇总)
| Formula Number | Name | Formula |
|---|---|---|
| (3.1) | Deformation Gradient | F = Gradχ = ∂χ/∂X |
| (3.4) | Volume Ratio | dv = J dV |
| (3.5) | Nanson's Formula | ds = J F⁻ᵀdS |
| (3.9) | Engineering Strain | ε = ½(grad u + (grad u)ᵀ) |
| (3.10) | Right Cauchy-Green | C = FᵀF |
| (3.11) | Left Cauchy-Green | b = FFᵀ |
| (3.12) | Green-Lagrange Strain | E = ½(C - I) |
| (3.13) | Euler-Almansi Strain | e = ½(I - b⁻¹) |
| (3.19) | Cauchy Stress Symmetry | σ = σᵀ |
| (3.20) | Cauchy Stress Theorem | t = σn |
| (3.22) | Stress Invariants | I₁, I₂, I₃ |
| (3.27) | Stress Decomposition | σ = σ̄ - pI |
| (3.28) | von Mises Stress | σₘ = √(-3J₂) |
| (3.31) | First Piola-Kirchhoff | P = JσF⁻ᵀ |
| (3.32) | Second Piola-Kirchhoff | S = JF⁻¹σF⁻ᵀ |
| (3.34) | Velocity Gradient | l = gradv = ḞF⁻¹ |
| (3.35) | Rate of Deformation | d = ½(l + lᵀ) |
| (3.36) | Spin Tensor | w = ½(l - lᵀ) |
| (3.38) | Rate of Volume Change | J̇ = J divv |
| (3.41) | Deformation Power | ψ̇ = S : Ċ/2 |
| (3.46) | Moduli Relation | E = 2(1+ν)G |
| (3.49) | Hooke's Law (Voigt) | ε = (1/E)gσ |
| (3.50) | Hooke's Law (inv) | σ = Ecε |
| (3.54) | S from Free Energy | S = 2∂ψ/∂C |
| (3.55) | Free Energy Split | ψ = ψᵢₛₒ + ψᵥₒₗ |
| (3.59) | Incomp. S (hyperelastic) | S = 2Dev[∂ψᵢₛₒ/∂C] - κC⁻¹ |
| (3.61) | Newtonian Fluid | σ = 2ηd - pI |
| (3.64) | Maxwell Element ODE | ε̇ = σ̇/E + σ/η |
| (3.100) | Mass Conservation | Dm/Dt = 0 |
| (3.102) | Continuity Eq. | ∂ρ/∂t + ρdivv = 0 |
| (3.107) | Cauchy Momentum | ρDv/Dt = divσ + bƒ |
| (3.115) | Navier-Stokes | ρDv/Dt = ηdiv(gradv) - gradp + bƒ |
| (3.124) | Clausius-Duhem | -ψ̇ + σ : d ≥ 0 |
| (3.127) | Coleman-Noll P | P = ∂ψ/∂F |
| (3.129) | Coleman-Noll S | S = 2∂ψ/∂C |
| (3.131) | Incompressible S | S = 2∂ψᵢₛₒ/∂C - κC⁻¹ |
| (3.148) | PVW (small def.) | ∫σ:δε = ∫bƒ·δu + ∫t·δu |
| (3.150) | PVW (material) | ∫S:δE = ∫Bƒ·δu + ∫T·δu |
| (3.151) | Isotropic Damage | σ = (1-D)σᵉᶠᶠ |
| (3.155) | J-Integral | J = ∫(ψdx₂ - t·∂u/∂x₁ds) |
| (3.156) | Cohesive TSL | T = ∂ψc/∂uᵈ, D = -∂ψc/∂ζ |
Notes compiled: 2022-Vascular-Biomechanics-Gasser Chapter 3